© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a226 K11a228

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X12,4,13,3 X18,6,19,5 X22,8,1,7 X16,10,17,9 X2,12,3,11 X8,14,9,13 X20,16,21,15 X10,18,11,17 X6,20,7,19 X14,22,15,21

Gauss Code:

{1, -6, 2, -1, 3, -10, 4, -7, 5, -9, 6, -2, 7, -11, 8, -5, 9, -3, 10, -8, 11, -4}

DT (Dowker-Thistlethwaite) Code:

4 12 18 22 16 2 8 20 10 6 14

Alexander Polynomial:

5t-3 - 17t-2 + 31t-1 - 37 + 31t - 17t2 + 5t3

Conway Polynomial:

1 + 8z2 + 13z4 + 5z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{143, 6}

Jones Polynomial:

q3 - 3q4 + 9q5 - 14q6 + 20q7 - 23q8 + 23q9 - 21q10 + 15q11 - 9q12 + 4q13 - q14

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q10 - 2q12 + 4q14 - q16 + q18 + 5q20 - 3q22 + 5q24 - 3q26 - q28 - 5q32 + 3q34 - 2q36 + 2q40 - q42

HOMFLY-PT Polynomial:

a-12 - a-12z2 - a-12z4 - 5a-10 - 6a-10z2 + a-10z6 + 4a-8 + 12a-8z2 + 11a-8z4 + 3a-8z6 + a-6 + 3a-6z2 + 3a-6z4 + a-6z6

Kauffman Polynomial:

- a-17z3 + a-17z5 + a-16z2 - 5a-16z4 + 4a-16z6 - 2a-15z + 7a-15z3 - 12a-15z5 + 8a-15z7 - 2a-14z2 + 9a-14z4 - 15a-14z6 + 10a-14z8 - 4a-13z + 15a-13z3 - 15a-13z5 - a-13z7 + 7a-13z9 + a-12 - 4a-12z2 + 25a-12z4 - 38a-12z6 + 16a-12z8 + 2a-12z10 - 10a-11z + 24a-11z3 - 14a-11z5 - 13a-11z7 + 12a-11z9 + 5a-10 - 16a-10z2 + 29a-10z4 - 34a-10z6 + 12a-10z8 + 2a-10z10 - 8a-9z + 20a-9z3 - 18a-9z5 - a-9z7 + 5a-9z9 + 4a-8 - 12a-8z2 + 15a-8z4 - 14a-8z6 + 6a-8z8 + 3a-7z3 - 6a-7z5 + 3a-7z7 - a-6 + 3a-6z2 - 3a-6z4 + a-6z6

V2 and V3, the type 2 and 3 Vassiliev invariants:

{8, 21}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=6 is the signature of 11227. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 r = 10 r = 11 j = 29                       1 j = 27                     3   j = 25                   6 1   j = 23                 9 3     j = 21               12 6       j = 19             11 9         j = 17           12 12           j = 15         8 11             j = 13       6 12               j = 11     3 8                 j = 9     6                   j = 7 1 3                     j = 5 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 227]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 227]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[18, 6, 19, 5], X[22, 8, 1, 7], > X[16, 10, 17, 9], X[2, 12, 3, 11], X[8, 14, 9, 13], X[20, 16, 21, 15], > X[10, 18, 11, 17], X[6, 20, 7, 19], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 227]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -7, 5, -9, 6, -2, 7, -11, 8, -5, 9, -3, 10, > -8, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 227]]
Out[5]=	DTCode[4, 12, 18, 22, 16, 2, 8, 20, 10, 6, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 227]][t]
Out[6]=	5 17 31 2 3 -37 + -- - -- + -- + 31 t - 17 t + 5 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 227]][z]
Out[7]=	2 4 6 1 + 8 z + 13 z + 5 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 227]}
In[9]:=	{KnotDet[Knot[11, Alternating, 227]], KnotSignature[Knot[11, Alternating, 227]]}
Out[9]=	{143, 6}
In[10]:=	J=Jones[Knot[11, Alternating, 227]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 12 q - 3 q + 9 q - 14 q + 20 q - 23 q + 23 q - 21 q + 15 q - 9 q + 13 14 > 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 227]}
In[12]:=	A2Invariant[Knot[11, Alternating, 227]][q]
Out[12]=	10 12 14 16 18 20 22 24 26 28 32 q - 2 q + 4 q - q + q + 5 q - 3 q + 5 q - 3 q - q - 5 q + 34 36 40 42 > 3 q - 2 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 227]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -12 5 4 -6 z 6 z 12 z 3 z z 11 z 3 z z a - --- + -- + a - --- - ---- + ----- + ---- - --- + ----- + ---- + --- + 10 8 12 10 8 6 12 8 6 10 a a a a a a a a a a 6 6 3 z z > ---- + -- 8 6 a a
In[14]:=	Kauffman[Knot[11, Alternating, 227]][a, z]
Out[14]=	2 2 2 2 -12 5 4 -6 2 z 4 z 10 z 8 z z 2 z 4 z 16 z a + --- + -- - a - --- - --- - ---- - --- + --- - ---- - ---- - ----- - 10 8 15 13 11 9 16 14 12 10 a a a a a a a a a a 2 2 3 3 3 3 3 3 4 4 12 z 3 z z 7 z 15 z 24 z 20 z 3 z 5 z 9 z > ----- + ---- - --- + ---- + ----- + ----- + ----- + ---- - ---- + ---- + 8 6 17 15 13 11 9 7 16 14 a a a a a a a a a a 4 4 4 4 5 5 5 5 5 5 25 z 29 z 15 z 3 z z 12 z 15 z 14 z 18 z 6 z > ----- + ----- + ----- - ---- + --- - ----- - ----- - ----- - ----- - ---- + 12 10 8 6 17 15 13 11 9 7 a a a a a a a a a a 6 6 6 6 6 6 7 7 7 7 4 z 15 z 38 z 34 z 14 z z 8 z z 13 z z > ---- - ----- - ----- - ----- - ----- + -- + ---- - --- - ----- - -- + 16 14 12 10 8 6 15 13 11 9 a a a a a a a a a a 7 8 8 8 8 9 9 9 10 10 3 z 10 z 16 z 12 z 6 z 7 z 12 z 5 z 2 z 2 z > ---- + ----- + ----- + ----- + ---- + ---- + ----- + ---- + ----- + ----- 7 14 12 10 8 13 11 9 12 10 a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 227]], Vassiliev[3][Knot[11, Alternating, 227]]}
Out[15]=	{8, 21}
In[16]:=	Kh[Knot[11, Alternating, 227]][q, t]
Out[16]=	5 7 7 9 2 11 2 11 3 13 3 13 4 q + q + 3 q t + 6 q t + 3 q t + 8 q t + 6 q t + 12 q t + 15 4 15 5 17 5 17 6 19 6 19 7 > 8 q t + 11 q t + 12 q t + 12 q t + 11 q t + 9 q t + 21 7 21 8 23 8 23 9 25 9 25 10 > 12 q t + 6 q t + 9 q t + 3 q t + 6 q t + q t + 27 10 29 11 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a227

K11a226 K11a228